Samplinglib
Lean gate passed 2026-08-19T06:32:39.895922+00:00 · 7bcd37294df1
Reviewed teaching declaration

radialSmoothCutoff

CompiledPartial

A radial cutoff at scale R is obtained by applying a fixed smooth one-dimensional cutoff to norm(x)/R.

Plain-English statement

A radial cutoff at scale R is obtained by applying a fixed smooth one-dimensional cutoff to norm(x)/R.

Mathematical statement

chi_R(x) = smoothUnitCutoff(norm(x)/R).

Intuition

The family is one on an inner ball, zero outside a larger ball, and approaches one pointwise as R grows. It is the approximation device for replacing compactly supported integration by parts with a whole-space statement.

Conditions

  • The ambient space is normed.
  • Scale-dependent derivative results separately require R > 0.

Why these conditions cannot be dropped

  • Division by a nonpositive scale does not describe an expanding cutoff.
  • Nondifferentiability of the norm at zero must be handled by the plateau region, not ignored.

Proof route

  • Start from a fixed smooth unit cutoff on the real line.
  • Compose with the scaled norm.
  • Prove plateau, support, smoothness, derivative bound, compact support, and convergence as separate leaves.
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Lean statement

noncomputable def radialSmoothCutoff (R : ℝ) (x : E) : ℝ :=
  smoothUnitCutoff (‖x‖ / R)

/-- The radial cutoff is one on the closed ball of radius `R`. -/

Lean interface notes

  • The definition itself is total in R; meaningful analytic theorems state positivity hypotheses.
  • HasCompactSupport and Function.support are different APIs and have separate bridge lemmas.